Classical cumulants from the double tensor unshuffle Hopf algebra
Abstract
We realize classical cumulants and their fixed-point equations in the double tensor unshuffle Hopf algebra introduced by Ebrahimi-Fard and Patras, placing classical, free, Boolean, and monotone cumulants in a common algebraic framework. A concatenation perturbation of the functionals yields left and right half-shuffle equations on ordinary words without changing the coproduct or either half-coproduct. Both equations are equivalent to the classical moment--cumulant formula and uniquely determine the classical cumulants. We also obtain a Hopf-algebraic realization of the classical-to-free cumulant transform through convolution exponentiation and the left half-shuffle logarithm. A commutative diagram identifies its partition-level action with summation over non-crossing closure fibers, recovering Lehner's closure-fiber formula within the framework of the double tensor unshuffle Hopf algebra.
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BibTeX
@misc{gao2026classical,
title = {{Classical cumulants from the double tensor unshuffle Hopf algebra}},
author = {Xing Gao and Jingtao Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33382},
eprint = {2610.33382},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33382},
}
RIS
TY - UNPB
TI - Classical cumulants from the double tensor unshuffle Hopf algebra
AU - Xing Gao
AU - Jingtao Li
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33382
UR - https://arxiv.org/abs/2610.33382
AB - We realize classical cumulants and their fixed-point equations in the double tensor unshuffle Hopf algebra introduced by Ebrahimi-Fard and Patras, placing classical, free, Boolean, and monotone cumulants in a common algebraic framework. A concatenation perturbation of the functionals yields left and right half-shuffle equations on ordinary words without changing the coproduct or either half-coproduct. Both equations are equivalent to the classical moment--cumulant formula and uniquely determine the classical cumulants. We also obtain a Hopf-algebraic realization of the classical-to-free cumulant transform through convolution exponentiation and the left half-shuffle logarithm. A commutative diagram identifies its partition-level action with summation over non-crossing closure fibers, recovering Lehner's closure-fiber formula within the framework of the double tensor unshuffle Hopf algebra.
ER -